Solving For An Integer A Mathematical Word Problem Explained
Introduction
In the realm of mathematics, problem-solving stands as a cornerstone of intellectual exploration. Word problems, in particular, serve as a bridge connecting abstract concepts to tangible scenarios, fostering critical thinking and analytical skills. The problem at hand, "A positive integer increased by 5 is equal to 36 times the multiplicative inverse (reciprocal) of the integer. Find the integer," presents a fascinating challenge that invites us to delve into the world of algebraic thinking. This article will dissect this problem, not just to arrive at the solution, but to illuminate the underlying principles and strategies that empower us to tackle similar mathematical puzzles. We will explore the process of translating verbal statements into algebraic equations, manipulating these equations to isolate the unknown, and ultimately, interpreting the solution within the context of the original problem. This journey through a single mathematical problem will reveal the broader landscape of algebraic thinking, emphasizing its power and elegance in deciphering the mysteries of numbers.
Deconstructing the Problem Statement
At the heart of every word problem lies a narrative, a story woven with mathematical threads. To decipher this story, we must first meticulously dissect the problem statement, identifying the key elements and their relationships. In our case, the central character is "a positive integer," an unknown quantity that we seek to unveil. This unknown is our protagonist, and we will represent it algebraically with the variable x. The narrative unfolds as this integer undergoes a series of transformations. First, it is "increased by 5," a simple addition that translates to x + 5. Then, we encounter the concept of a "multiplicative inverse (reciprocal)," a term that might initially seem daunting but simply refers to 1 divided by the number. The reciprocal of our integer x is thus 1/x. The problem statement then introduces a multiplicative factor, stating that the result of increasing the integer by 5 is equal to "36 times the multiplicative inverse." This translates to 36 * (1/x), or 36/x. The final piece of the puzzle is the equality, the bridge that connects the two sides of our equation. The statement asserts that x + 5 is equal to 36/x, allowing us to construct the algebraic equation that will guide us to the solution. This careful deconstruction of the problem statement is the crucial first step, transforming a verbal narrative into a precise mathematical expression.
From Words to Equations The Algebraic Translation
Translating word problems into algebraic equations is akin to deciphering a secret code, transforming the language of words into the language of symbols. This process requires careful attention to detail, ensuring that each phrase and relationship is accurately represented in mathematical form. In our problem, the phrase "a positive integer increased by 5" directly translates to the algebraic expression x + 5, where x represents the unknown integer. The concept of a "multiplicative inverse (reciprocal)" becomes 1/x, capturing the essence of inverting the number. The phrase "36 times the multiplicative inverse" then morphs into 36 * (1/x), which can be simplified to 36/x. The crucial word "is equal to" serves as the linchpin, the equals sign (=) that connects the two sides of our equation. By piecing together these translated components, we arrive at the algebraic equation x + 5 = 36/x. This equation encapsulates the essence of the problem statement, providing a mathematical framework for solving the unknown integer. The art of algebraic translation lies in the ability to identify these key phrases and relationships, transforming the narrative into a concise and solvable equation. This equation is the map that will guide us through the mathematical landscape, leading us to the hidden treasure of the integer we seek.
Solving the Equation A Step-by-Step Approach
With our algebraic equation x + 5 = 36/x in hand, we embark on the journey of solving for the unknown integer x. This process involves a series of strategic manipulations, each designed to isolate x on one side of the equation. The first step is to eliminate the fraction, a common hurdle in algebraic equations. We achieve this by multiplying both sides of the equation by x, resulting in x(x + 5) = 36. This crucial step transforms the equation into a more manageable form, removing the denominator and paving the way for further simplification. Next, we distribute the x on the left side, expanding the expression to x² + 5x = 36. This reveals a quadratic equation, a type of equation that involves a squared term. To solve quadratic equations, we typically rearrange them into the standard form ax² + bx + c = 0. Subtracting 36 from both sides of our equation, we obtain x² + 5x - 36 = 0. Now, we have a quadratic equation in standard form, ready to be tackled. The most common methods for solving quadratic equations are factoring, using the quadratic formula, or completing the square. In this case, the equation is readily factorable. We seek two numbers that multiply to -36 and add to 5. These numbers are 9 and -4. Thus, we factor the quadratic as (x + 9)(x - 4) = 0. This factored form reveals the solutions directly. For the product of two factors to be zero, at least one of the factors must be zero. Therefore, either x + 9 = 0 or x - 4 = 0. Solving these simple linear equations, we find two potential solutions: x = -9 and x = 4. However, we must remember the original problem's constraint: we are seeking a positive integer. Therefore, the solution x = -9 is extraneous, a mathematical red herring. The only valid solution is x = 4. This step-by-step approach, from eliminating the fraction to factoring the quadratic, demonstrates the power of algebraic manipulation in unraveling mathematical equations.
Verifying the Solution The Test of Truth
In the realm of mathematics, finding a solution is not the final destination; it is merely a milestone. The true test of a solution lies in its verification, ensuring that it satisfies the original conditions of the problem. In our case, we have arrived at the potential solution x = 4 for the positive integer. To verify this, we must substitute this value back into the original equation, x + 5 = 36/x, and check if the equality holds. Substituting x = 4 into the left side of the equation, we get 4 + 5 = 9. Substituting x = 4 into the right side of the equation, we get 36/4 = 9. Since both sides of the equation evaluate to 9, the equality holds. This confirms that x = 4 is indeed a valid solution. But verification goes beyond just satisfying the equation. We must also ensure that the solution aligns with the original problem statement's constraints. The problem specified that we are seeking a positive integer, and 4 certainly fits this criterion. Thus, our solution passes both the algebraic test and the contextual test, solidifying its validity. This process of verification is a crucial safeguard against errors, ensuring that our mathematical journey has led us to the correct destination. It is the final flourish, the stamp of authenticity on our solution.
Interpreting the Result The Answer in Context
Having verified our solution, x = 4, we arrive at the final stage of problem-solving: interpreting the result within the context of the original problem. This is where the abstract mathematics transforms back into a tangible answer, a meaningful response to the question posed. The problem asked us to find a positive integer that, when increased by 5, is equal to 36 times its multiplicative inverse. Our solution, x = 4, represents this integer. Therefore, the answer to the problem is that the integer is 4. This simple statement encapsulates the entire mathematical journey, from the initial deconstruction of the problem to the final verification of the solution. But the interpretation goes beyond just stating the answer. It involves connecting the mathematical result back to the narrative of the problem, ensuring that the solution makes sense in the real-world context (even if the context is a purely mathematical one). In this case, we can say that the positive integer 4 has the unique property that when you add 5 to it, you get the same result as multiplying its reciprocal (1/4) by 36. This interpretation provides a deeper understanding of the problem and its solution, highlighting the interconnectedness of mathematical concepts. It is the final brushstroke, completing the masterpiece of problem-solving.
Embracing Algebraic Thinking The Broader Perspective
Our journey through this single word problem has illuminated the core principles of algebraic thinking, a powerful tool for navigating the world of mathematics and beyond. Algebraic thinking is not just about manipulating equations; it is about developing a way of thinking, a systematic approach to problem-solving. It involves the ability to translate verbal statements into mathematical expressions, to identify patterns and relationships, and to use symbols and variables to represent unknown quantities. In this problem, we saw how algebraic thinking allowed us to transform a wordy description into a concise equation, x + 5 = 36/x. We then used algebraic techniques to solve this equation, revealing the hidden integer. But the benefits of algebraic thinking extend far beyond solving specific problems. It fosters critical thinking, logical reasoning, and analytical skills, all of which are valuable assets in any field. By embracing algebraic thinking, we empower ourselves to tackle complex challenges, to break down problems into smaller, manageable steps, and to arrive at logical solutions. It is a way of thinking that encourages exploration, experimentation, and a willingness to grapple with the unknown. As we continue our mathematical journey, the principles of algebraic thinking will serve as our compass, guiding us through the intricate landscape of numbers and equations.
Conclusion
In conclusion, the problem of finding a positive integer that, when increased by 5, equals 36 times its multiplicative inverse serves as a compelling example of the power of algebraic thinking. We have seen how the seemingly simple act of translating words into equations can unlock hidden solutions, and how the systematic manipulation of these equations can lead us to the truth. The journey through this problem has highlighted the key elements of algebraic thinking: deconstructing the problem statement, translating words into equations, solving equations using algebraic techniques, verifying the solution, and interpreting the result in context. These skills are not just applicable to mathematical problems; they are transferable to a wide range of challenges in life. By embracing algebraic thinking, we cultivate a mindset of problem-solving, a way of approaching the world with curiosity, logic, and a willingness to explore the unknown. This article has not just solved a problem; it has illuminated a path, a path that leads to a deeper understanding of mathematics and its power to illuminate the world around us.
